The prior answers are good, but I wanted to point out a couple of other things.
First, as one of the answers noted, the "grill" was not a trophy sought by Hank, but rather something that was given to him by his colleagues.   Second, it seemed to me that Tuco was actually killed by the gunshot Tuco suffered during the strugggle with Jesse, fired by Jesse from Tuco's own gun before Hank arrived on the scene.  Although Tuco was still alive during the gunfight with Hank, it was not clear that Hank even hit Tuco and Hank likely would have died had Tuco not been shot prior to Hank showing up.  I think Hank felt that the trophy was not deserved.